We study the nonlinear fractional equation (−∆)u = f(u) in R, for all fractions 0 < s < 1 and all nonlinearities f . For every fractional power s ∈ (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional ...

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: we are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e) ≤ g(e) for each hyperedge e. The task is to find a minimum cost basis which ...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the well-known definition of a local embedding of an algebraic system A in a class K of algebraic systems. According to this definition A is locally embedded in K iff it is a subsystem...

Let ηi, i = 1, . . . , n be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n elements v1, . . . , vn of an additive group G, we define the concentration probability of V as ρ(V ) := sup v∈G P(η1v1 + . . . ηnvn = v). An old result of Erdős and Moser asserts that if vi are distinct real numbers then ρ(V ) is O(n− 3 2 logn). This bound was then refine...

We construct a transcendental entire f : C ? such that (1) has bounded singular set, (2) wandering domain, and (3) each value of escapes to infinity under iteration by f.

We show that bounded solutions of the quasilinear elliptic equation \(\Delta_{p(x)} u=g+div(\textbf{F})\) are locally Hölder continuous provided functions \(g\) and \(\textbf{F}\) in suitable Lebesgue spaces.

Let $Omega_X$ be a bounded, circular and strictly convex domain of a Banach space $X$ and $mathcal{H}(Omega_X)$ denote the space of all holomorphic functions defined on $Omega_X$. The growth space $mathcal{A}^omega(Omega_X)$ is the space of all $finmathcal{H}(Omega_X)$ for which $$|f(x)|leqslant C omega(r_{Omega_X}(x)),quad xin Omega_X,$$ for some constant $C>0$, whenever $r_{Omega_X}$ is the M...

(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0)...